1	Huffman Encoding Week 2 Min-Yen KAN (Self-Study Module)
2	Fixed and variable bit widths To encode English text, we need 26 lower case letters, 26 upper case letters, and a handful of punctuation We can get by with 64 characters (6 bits) in all Each character is therefore 6 bits wide We can do better, provided: Some characters are more frequent than others Characters may be different bit widths, so that for example, e use only one or two bits, while x uses several We have a way of decoding the bit stream Must tell where each character begins and ends
3	Example Huffman encoding A = 0 B = 100 C = 1010 D = 1011 R = 11 ABRACADABRA = 01001101010010110100110 This is eleven letters in 23 bits A fixed-width encoding would require 3 bits for five different letters, or 33 bits for 11 letters Notice that the encoded bit string can be decoded!
4	Why it works In this example, A was the most common letter In ABRACADABRA: 5 As code for A is 1 bit long 2 Rs code for R is 2 bits long 2 Bs code for B is 3 bits long 1 C code for C is 4 bits long 1 D code for D is 4 bits long
5	Creating a Huffman encoding For each encoding unit (letter, in this example), associate a frequency (number of times it occurs) You can also use a percentage or a probability Create a binary tree whose children are the encoding units with the smallest frequencies The frequency of the root is the sum of the frequencies of the leaves Repeat this procedure until all the encoding units are in the binary tree
6	Example, step I Assume that relative frequencies are: A: 40 B: 20 C: 10 D: 10 R: 20 Smallest number are 10 and 10 (C and D), connect those
7	Example, step II C and D have already been used, and the new node above them (call it C+D) has value 20 The smallest values are B, C+D, and R, all of which have value 20 Connect any two of these
8	Example, step III The smallest values is R, while A and B+C+D all have value 40 Connect R to either of the others
9	Example, step IV Connect the final two nodes
10	Example, step V Assign 0 to left branches, 1 to right branches Each encoding is a path from the root A = 0 B = 100 C = 1010 D = 1011 R = 11 Each path terminates at a leaf Do you see why encoded strings are decodable?
11	Unique prefix property A = 0 B = 100 C = 1010 D = 1011 R = 11 No bit string is a prefix of any other bit string For example, if we added E=01, then A (0) would be a prefix of E Similarly, if we added F=10, then it would be a prefix of three other encodings (B=100, C=1010, and D=1011) The unique prefix property holds because, in a binary tree, a leaf is not on a path to any other node
12	Practical considerations Is encoding practical for long texts or short ones? Short: impractical To decode it, you would need the code table The code table is bigger than the message Long: practical The encoded string is large relative to the code table